## Bulletin of the Belgian Mathematical Society - Simon Stevin

### On nuclearity of the algebra of adjointable operators

#### Abstract

We study nuclearity of the $C^*$-algebra $\mathbb B(\mathcal E)$ of adjointable operators on a full Hilbert $C^*$-module $\mathcal E$ over a $C^*$-algebra $\mathcal A$. When $\mathcal A$ is a von Neumann algebra and $\mathcal E$ is full and self dual, we show that $\mathbb B(\mathcal E)$ is nuclear if and only if $\mathcal A$ is nuclear and $\mathcal E$ is finitely generated. In particular, when $\mathcal A$ is a factor, then nuclearity of $\mathbb B(\mathcal E)$ implies that $\mathcal E$, $\mathcal A$ and $\mathbb B(\mathcal E)$ are finite dimensional.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 22, Number 3 (2015), 423-427.

Dates
First available in Project Euclid: 16 September 2015

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1442364589

Digital Object Identifier
doi:10.36045/bbms/1442364589

Mathematical Reviews number (MathSciNet)
MR3396993

Zentralblatt MATH identifier
1334.46042

#### Citation

Amini, Massoud; Asadi, Mohammad B. On nuclearity of the algebra of adjointable operators. Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 3, 423--427. doi:10.36045/bbms/1442364589. https://projecteuclid.org/euclid.bbms/1442364589